Compound Annual Growth Rate (CAGR)
Introduction
The Compound Annual Growth Rate (CAGR) is a metric that is used to measure the mean annual growth rate of an investment over a specified time period longer than one year. CAGR is a useful measure because it represents one of the most accurate ways to calculate and determine returns for anything that can rise or fall in value over time. Unlike average annual returns, CAGR represents one rate that, if compounded over a period of years, would yield the investment’s ending value.
Formula and Calculation
The formula for CAGR is:
[ \text{CAGR} = \left( \frac{EV}{BV} \right)^{\frac{1}{n}} - 1 ]
Where:
Let’s break it down with an example:
- Suppose you have an investment that was worth $1,000 at the beginning and $2,000 after 5 years.
- The formula would be applied as follows:
[ \text{CAGR} = \left( \frac{2000}{1000} \right)^{\frac{1}{5}} - 1 ] [ \text{CAGR} = (2)^{0.2} - 1 ] [ \text{CAGR} \approx 0.1487 \text{ or } 14.87\% ]
Thus, the investment has grown at an annual compound growth rate of approximately 14.87%.
Importance in Algorthmic Trading
In algorithmic trading, CAGR is essential for the following reasons:
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Performance Measurement: It provides a single annualized rate that shows the consistent growth of an investment portfolio over time, even if the growth happened at an irregular rate year over year.
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Portfolio Comparison: Traders can use CAGR to compare the performance of different portfolios or investment strategies over the same period because it normalizes the growth rates.
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Risk Evaluation: By comparing CAGR with the volatility of the investment (measured in terms of the standard deviation of returns), traders can evaluate the risk-adjusted performance of a strategy.
Practical Applications
Portfolio “A” vs. Portfolio “B”
Consider two portfolios over a 3-year period:
- Portfolio “A” grows from $10,000 to $13,000.
- Portfolio “B” fluctuates wildly but grows from $10,000 to $12,500.
To decide which portfolio performed better in terms of steady growth, we calculate the respective CAGRs:
Portfolio “A”:
[ \text{CAGR} = \left( \frac{13000}{10000} \right)^{\frac{1}{3}} - 1 ] [ \text{CAGR} = (1.3)^{0.3333} - 1 ] [ \text{CAGR} \approx 0.0914 \text{ or } 9.14\% ]
Portfolio “B”:
[ \text{CAGR} = \left( \frac{12500}{10000} \right)^{\frac{1}{3}} - 1 ] [ \text{CAGR} = (1.25)^{0.3333} - 1 ] [ \text{CAGR} \approx 0.0764 \text{ or } 7.64\% ]
Even though Portfolio “B” had significant upticks and downticks, Portfolio “A” demonstrated more consistent growth as indicated by a higher CAGR.
Comparison to Other Metrics
Average Annual Return
Unlike the average annual return, CAGR smoothes out the effects of volatility. For instance, an asset that gains 100% in one year and loses 50% the next year has an average annual return of 25% over those two years. However, its CAGR illustrates a different reality:
[ \text{CAGR} = \left( \frac{1.50}{1.00} \right)^{\frac{1}{2}} - 1 ] [ \text{CAGR} = (1.50)^{0.5} - 1 ] [ \text{CAGR} \approx 0.225 \text{ or } 22.5\% ]
Total Return
Total return refers to the total gain or loss on an investment, including dividends, interest, and capital gains. Unlike CAGR, it does not annualize the returns, making CAGR a preferable metric for comparing investments of different lengths.
Real-World Usage
Investors
Private and institutional investors use CAGR to gauge the performance of investments like stocks, bonds, and mutual funds. Websites like Morningstar often use CAGR when showcasing the historical performance of investment funds.
Financial Analysts
Analysts use CAGR in their financial models to predict future earnings and to evaluate the historical performance of a company. This can be seen in Analyst reports from companies like Goldman Sachs.
Businesses
Businesses use CAGR to monitor and report the growth of key performance indicators (KPIs) such as revenue, profits, and user base. Companies like Amazon display such data in their annual reports to showcase performance growth over time.
Limitations of CAGR
Despite its usefulness, CAGR has some limitations:
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Does Not Reflect Volatility: CAGR smoothes returns to show an average rate of growth, thereby masking the volatility an investment might experience from year to year.
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Past Performance: It’s backward-looking and may not necessarily predict future performance.
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Single Metric: It should not be used in isolation but rather combined with other metrics like standard deviation, Sharpe ratio, and maximum drawdown to get a comprehensive view.
Conclusion
CAGR provides a clear and straightforward metric for comparing the annual growth rates of investments over time. It’s a critical tool for traders, investors, analysts, and businesses to measure and compare performance accurately. Its main advantage is its ability to normalize the growth rates of different investments, providing a ‘smoother’ perspective on performance. However, users should be aware of its limitations and use it in conjunction with other metrics for a well-rounded analysis.